The Cusum Test with Ols Residuals

Abstract

We show that the CUSUM test of the stability over time of the coefficients of a linear regression model, which is usually based on recursive residuals, can also be applied to ordinary least squares residuals. We derive the limiting null distribution of the resulting test and compare its local power to that of the standard procedure. It turns out that neither version is uniformly superior to the other. henceforth BDE) on the CUSUM and CUSUM of squares tests for the constancy over time of the coefficients of a linear regression model. Both tests are based on recursive residuals, which are independent N(O, o.2) under Ho, and therefore ideal ingredients for all types of tests. While the CUSUM of squares test has been generalized to ordinary least squares (OLS) residuals (McCabe and Harrison (1980)), a generalization of the CUSUM test to OLS residuals, which are dependent and heteroskedastic even under Ho, has not yet been made except for a very special case (MacNeill (1978)). In fact, both BDE (1975, p. 151) and McCabe and Harrison (1980, p. 142) argue that a least squares variant of the CUSUM test poses intractable problems, due to an alleged difficulty in assessing the significance of the departure of the cumulated OLS residuals from their mean value zero. Below we show that it is no more difficult to derive the limiting distribution for a CUSUM test based on OLS residuals than for a CUSUM test based on recursive residuals. While the CUSUMs of the recursive residuals, properly standardized, tend in distribution to a standard Wiener process (Sen (1982); Krimer, Ploberger, and Alt (1988), henceforth KPA), we show that the OLS- based CUSUMs tend in distribution to a Brownian bridge (or tied-down- Brownian-motion; see Billingsley (1968, p. 64)). This generalizes MacNeill's (1978) polynomial regression result to arbitrary regressor sequences. We also demonstrate that the resulting CUSUM test has higher (local) power for certain types of structural change than the one based on recursive residuals, though neither variant is uniformly superior to the other.